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```CHAP, ill]                FERMAT'S AND WILSON'S THEOREMS.                        67
the sign being + or — according as &2=a (mod p) has or has not integral
solutions  (Euler's  criterion).   Squaring, we obtain Fermat's theorem.
Finally,, Dirichlet rediscovered the proof by Ivory.33   [Cf. Moreau.123] J. Binet41 also rediscovered the proof by Ivory.33 A. Cauchy42 gave a proof analogous to that by Euler.10 An anonymous writer43 proved that if n is a prune the binomial coeffi-
cient (n — I)*, has the residue ( — 1)* modulo n, so that
modulo n.   Thus Fermat's theorem follows by induction on # as in the proof by Euler.12
V. Bouniakowsky44 gave a proof of Euler's theorem similar to that by Laplace.23 If a=*=b is divisible by a prime p, a^^^b^'1 is divisible by pn, provided p> 2 when the sign is plus. Hence if p, p', . . . are distinct primes, a'=*= V is divisible by N=pnpfn' . . . , where t = pn"lpfn'~l . . . , if a=*= b is divisible by ppf - - • i provided the p's are > 2 if the sign is plus. Replace a by its (p— l)th power and 6 by 1 and use Fermat's theorem; we see that a*— 1 is divisible by N if e=4>(N*). The same result gives a generalization of Wilson's theorem6
He gave (ibid., 563-4) Gauss'30 proof of Wilson's theorem.
J. A. Grunert45 used the known fact that, if Q<k<p, then k, 2k, . . ., (p~l)k are congruent to 1, 2,. . ., p— 1 in some order modulo p, a prime, to show that kx=l (mod p) has a unique root x. Wilson's theorem then follows as by Gauss. If (ibid., p. 1095) we square Gauss' formula,35 we get Fermat's theorem.
Giovanni de Paoli46 proved Fermat's and Euler 's theorems.   In
where p is a prime, Sx is an integer.   Change x to a; — 1, . . . , 2, 1 and add the resulting equations.   Thus
Replace x by xm, divide by xm and set y=xT~1.   Thus
ym - 1 = PXa,          Xm -as,-/*" = integer.
Replace m by 2m, . . . , (p — l)m, add the resulting equations, and set Ym = 1 +Xm+X2m+ . . . +X(^m.   Thus
y""-l=p(ym-l)Ym=P2XmYm.
"Jour, de l'6cole polytechnique, 20, 1831, 291 (read 1827).   Cauchy, Comptes Rendua Paris,
12, 1841, 813, ascribed the proof to Binet.
"Exer. de math., 4, 1829, 221; Oeuvres, (2), 9, 263.    R6sum6 analyt., Turin, 1, 1833, 10. "Jour, fur Math., 6, 1830, 100-6.
"MSm. Ac. Sc. St. P<§tersbourg, Sc. Math. Phys. et Nat., (6), 1, 1831, 139 (read Apr. 1, 1829). «Klugel's Math. Worterbuch, 5, 1831, 1076-9. "Opuscoli Matematici e Fieici di Diversi Autori, Milano, 1, 1832, 262-272.```